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Engineering Calculus Notes 184

# Engineering Calculus Notes 184 - t −→ cos t −→ k t...

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172 CHAPTER 2. CURVES (j) ( x 1 ,y 1 ,z 1 ) = (1 , 2 , 3) , ( x n +1 ,y n +1 ,z n +1 ) = ( x n + 1 2 y n ,y n + 1 2 z n , 1 2 z n ) 2. An accumulation point of a sequence { −→ p i } of points is any limit point of any subsequence. Find all the accumulation points of each sequence below. (a) parenleftbigg 1 n , ( 1) n n n + 1 parenrightbigg (b) parenleftbigg n n + 1 cos n, n n + 1 sin n parenrightbigg (c) parenleftbigg n n + 1 , ( 1) n n 2 n + 1 , ( 1) n 2 n n + 1 parenrightbigg (d) parenleftbigg n n + 1 cos 2 , n n + 1 sin 2 , 2 n n + 1 parenrightbigg 3. For each vector-valued function −→ p ( t ) and time t = t 0 below, find the linearization T t 0 −→ p ( t ). (a) −→ p ( t ) = ( t,t 2 ), t = 1 (b) −→ p ( t ) = t 2 −→ ı t 3 −→ , t = 2 (c) −→ p ( t ) = (sin t, cos t ), t = 4 π 3 (d) −→ p ( t ) = (2 t + 1) −→ ı + (3 t 2 2) −→ , t = 2 (e) −→ p ( t ) = (sin t,
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Unformatted text preview: t ) −→ + cos t ) −→ k , t = π 2 Theory problems: 4. Prove the triangle inequality dist( P,Q ) ≤ dist( P,R ) + dist( R,Q ) (a) in R 2 ; (b) in R 3 . ( Hint: Replace each distance with its de±nition. Square both sides of the inequality and expand, cancelling terms that appear on both sides, and then rearrange so that the single square root is on on one side; then square again and move all terms to the same side of the...
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